protein folding dynamics and more
DESCRIPTION
Protein folding dynamics and more. Chi-Lun Lee ( 李紀倫 ). Department of Physics National Central University. For a single domain globular protein (~100 amid acid residues), its diameter ~ several nanometers and molecular mass ~ 10000 daltons (compact structure). Introduction. - PowerPoint PPT PresentationTRANSCRIPT
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Protein folding dynamics and more
Chi-Lun Lee (李紀倫 )
Department of Physics
National Central University
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For a single domain globular protein (~100 amid acid residues), its diameter ~ several nanometers and molecular mass ~ 10000 daltons (compact structure)
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Introduction
• N = 100 # of amino acid residues (for a single domain pro
tein)
• = 10 # of allowed conformations for each amino acid re
sidue
• For each time only one amino acid residue is allowed to c
hange its state
• A single configuration is connected to N = 1000 other co
nfigurations
Modeling for folding kinetics
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Concepts from chemical reactions
Transition state theory
F
Reaction coordinate
Unfolded
Transition state
Folded
F*
Arrhenius relation : kAB ~ exp(-F*/T)
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foldedunfolded
(order parameter)
For complex kinetics, the stories can be much more complicated
Statistical energy landscape theory
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Energy surface may be rough at times…
• Traps from local minima
• Non-Arrenhius relation
• Non-exponential relaxation
• Glassy dynamics
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Peak in specific heat vs. T
c
T
Resemblance with first order transitions (nucleation)?
Cooperativity in folding
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• Defining an order parameter
• Specifying a network
• Assigning energy distribution P(E,)
• Projecting the network on the order parameter continuou
s time random walk (CTRW)
Theory : to build up and categorize an energy landscape
Generalized master equation
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Random energy model
i =
– 0 , when the ith residue is in its native state.
a Gaussian random variable with mean – and variance when the residue is non-native.
– 0 native
– non-native
Bryngelson and Wolynes, JPC 93, 6902(1989)
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Random energy model
•Another important assumption : random erergy approximation (energies for different configurations are uncorrelated)
•This assumption was speculated by the fact that one conformational change often results in the rearrangements of the whole polypeptide chain.
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Random energy model
•For a model protein with N0 native residues, E(N0) is a
Gaussian random variable with mean
and variance
order parameter
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Random energy model
Using a microcanonical ensemble analysis, one can derive expressions for the entropy and therefore the free energ
y of the system:
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Kinetics : Metropolis dynamics+CTRW
Transition rate between two conformations
Folding (or unfolding) kinetics can be treated as random
walks on the network (energy landscape) generated from
the random energy model
( R0 ~ 1 ns )
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Random walks on a network (Markovian)
One-dimensional CTRW (non-Markovian)
Two major ingredients for CTRW :
•Waiting time distribution function
•Jumping probabilities
after mapping on
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can be derived from statistics of the escape rate :
And can be derived from the
equilibrium condition
equilibrium distribution :
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probability density that at time a random w
alker is at
probability for a random w
alker to stay at for at least time
probability to jump from to ’ in one step after time
Let us define
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0 jump 1 jump
2 jumps
Therefore
or
Generalized Fokker-Planck equation
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Results : mean first passage time (MFPT)
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Results : second moments
Poisson
long-time relaxation
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Results : first passage time (FPT) distribution
0 < < 1
Lévy distribution
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Power-law exponents for the FPT distribution
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Locating the folding transition
folding transition
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cf. simulations (Kaya and Chan, JMB 315, 899 (2002))
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Results : a dynamic ‘phase diagram’
(power-law decay)
(exponential decay)
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A fantasy from the protein folding problem…
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A ‘toy’ model : Rubik’s cube
3 x 3 x 3 cube : ~ 4x1019 configurations2 x 2 x 2 cube : 88179840 configurations
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Metropolis dynamics (on a 2 x 2 x 2 cube)
Transition rate between two conformations
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Monte Carlo simulations
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Energy : -(total # of patches coinciding with their central-face color)
0.E+00
2.E+06
4.E+06
6.E+06
8.E+06
1.E+07
1.E+07
1.E+07
2.E+07
2.E+07
2.E+07
E
Num
ber of
sta
tes
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-24
-20
-16
-12
-8
-4
0
0 2 4 6 8 10
T
E
0
5
10
15
20
25
30
35
0 2 4 6 8 10 12
T
Cv
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1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
0 5 10 15
Depth
Num
ber
of c
onfi
gura
tion
s
A possible order parameter : depth (minimal # of steps from the native state)
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-30
-25
-20
-15
-10
-5
0
0 2 4 6 8 10 12 14 16
Depth
E
Funnel-like energy landscape
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Free energy
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Energy fluctuations (T=1.3)
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• A strectched exponential relaxation
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Two timing in the ‘folding’ process : 1 , 2
Anomalous diffusion
Rolling along the order parameter
‘downhill’ : R1 >>1
‘uphill’ : R1 <<1
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Summary
• Random walks on a complex energy landscape statistic
al energy landscape theory (possibly non-Markovian)
• Local minima (misfolded states)
• Exponential nonexponential kinetics
• Nonexponential kinetics can happen even for a ‘downhill’
folding process (cf. experimental work by Gruebele et al.,
PNAS 96, 6031(1999))
Acknowledgment : Jin Wang, George Stell
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U
F
1 , 2
T
F
3 , 4
U
1 , 2
•If T is high (e.g., entropy associated with transition state ensemble is small) exponential kinetics likely
•If T is low or there is no T nonexponential kinetics
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short-time scale : exponential decay
long-time scale : power-law decay
Waiting time distribution function
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Results : diffusion parameter
Lee, Stell, and Wang, JCP 118, 959 (2003)